This problem has been around forever. This is a quote from a Standard Triumph
service bulletin dated March 1961:
"SUBJECT: GENERAL SERVICE HINTS - TR-3
Gearshift Lever Chatter
The simple corrective measure has been dealt with in a separate Bulletin which seems to have been overlooked by many service shops. Therefore, to briefly recapitulate, remove the gearshift lever knob and boot, remove the top cover bolt and cover together with the inner cover and spring. Slide down over the the gearshift lever a fairly heavy piece of rubber or neophrene (sic) tube 7/8" long and 5/8" inside diameter. Press it down until it comes in contact with the fulcrum ball. Reassemble and that is all that is necessary. The rubber or neophrene tube will act as a vibration damper in this position............" I did this to my 3 about 35K miles ago & haven't had a rattle since.
FRank